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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"0b864ffb22ad13ee8921d29d74606c3ba4fdc1e3e168e911c69277294c696d5f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           312,            266,             49;\n  1:          4752,           6552,           2793,             343;\n  2:         69120,         133056,          91728,           26068,            2401;\n  3:        974592,        2419200,        2328480,         1070160,          228095,           16807;\n  4:      13436928,       40932864,       50803200,        32598720,        11236680,         1915998,         117649;\n  5:     182145024,      658409472,     1002855168,       829785600,       399334320,       110119464,       15647317,         823543;\n  6:    2436562944,    10200121344,    18435465216,     18719963136,     11616998400,      4472544384,     1027781664,      125178536,       5764801;\n  7:   32248627200,   153503465472,   321303822336,    387144769536,    294839419392,    146374179840,    46961716032,     9250034976,     985780971,     40353607;\n  8:  423101988864,  2257403904000,  5372621291520,   7497089187840,   6775033466880,   4127751871488,  1707698764800,   469617160320,   80937806040,   7667185330,   282475249;\n  9: 5510645415936, 32578853142528, 86910050304000, 137897279815680, 144318966865920, 104335515389952, 52972815684096, 18784686412800, 4520065168080, 692467896120, 59037327041, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy (7*n+19+7*x)*P(n,x) = (84*n+312+(49*n+266)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 312 + 266*x + 49*x^2.","mathematica":"P[0, x_] := 312 + 266*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((84*n + 312 + (49*n + 266)*x + 49*x^2)*P[n - 1, x])/(7*n + 19 + 7*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 312 + 266*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((84*n + 312 + (49*n + 266)*x + 49*x^2)*v[i - 1])/(7*n + 19 + 7*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5cf6acfdac607d6d","steps":["D+1"],"edges":["e-f82bc0c28c0fdee7"],"text":"A038333 \u2192 D+1"},"headline":"A038333 \u2192 D+1","chain":null}